EXCHANGE 


H!l    18  I'M' 

THE  MEASUREMENT  OF  THE  COEFFICIENT 

OF  VISCOSITY  BY  MEANS  OF  THE 
FORCED  VIBRATIONS  OF  A  SPHERE 


BY 

GEORGE   FRANCIS  McEWEN 

A.B.,  Stanford  University,  1908 


A  THESIS 

Submitted  in  Partial  Fulfillment  of  the  Requirements  for  the  Degree  of 

Doctor  of  Philosophy  in  Physics  at 
The  Leland  Stanford  Junior  University,  1911 


mss  OP 

THE  NEW  ERA  PR'KTtNO  COMPANY 
LAftCASTCR.  PA 


THE  MEASUREMENT  OF  THE  COEFFICIENT 

OF  VISCOSITY  BY  MEANS  OF  THE 
FORCED  VIBRATIONS  OF  A  SPHERE 


BY 

GEORGE   FRANCIS  McEWEN 

A.B.,  Stanford  University,  1908 


A  THESIS 

Submitted  in  Partial  Fulfillment  of  the  Requirements  for  the  Degree  of 

Doctor  of  Philosophy  in  Physics  at 
The  Leland  Stanford  Junior  University,  1911 


PRCSS  OF 

THE  NEW  ERA  PP'NTINO  COMPANY 
LANCASTER.  PA 

IQTI 


(Reprinted  from  the  PHYSICAL  REVIEW,  Vol.  XXXIII.,  No.  6,  December,  1911.1 


THE  MEASUREMENT  OF  THE  FRICTIONAL  FORCE  EX- 
ERTED ON  A  SPHERE  BY  A  VISCOUS  FLUID,  WHEN  THE 
CENTER  OF  THE  SPHERE  PERFORMS  SMALL  PERIODIC 
OSCILLATIONS  ALONG  A  STRAIGHT  LINE. 

BY  GEORGE  F.  McEwEN. 

^HE  purpose  of  the  following  investigation  was  to  devise  a  method 
by  which  Stokes's  law  of  the  frictional  resistance  of  a  fluid  to 
the  motion  of  a  sphere  whose  center  performs  small  periodic  oscilla- 
tions could  be  applied  to  the  measurement  of  the  coefficient  of  viscosity 
of  fluids.  The  two  main  objects  were:  first,  to  obtain  as  close  an  agree- 
ment as  possible  between  the  actual  working  conditions  and  those  de- 
manded by  theory;  second,  to  devise  a  process  of  measuring  the  force 
acting  on  the  sphere,  even  for  fluids  having  a  very  large  coefficient  of 
viscosity. 

The  contents  of  this  paper  fall  under  the  following  five  heads: 

I.  The  effect  of  the  internal  friction  of  fluids  on  the  motion  of  pendu- 
lums, from  Sir  G.  G.  Stokes's  Math,  and  Phys.  Papers,  Cambridge,  1880 
and  1901,  Vols.  I.  and  III. 

II.  An  account  of  the  method  adopted  in  the  present  investigation 
to  overcome  the  difficulties  mentioned  by  Stokes,  and  to  more  nearly 
realize  in  the  experimental  work  the  ideal  condition  assumed  in  the 
theory  from  which  Stokes's  law  was  deduced. 

III.  Experimental  tests  of  the  present  method. 

IV.  Suggestions  for  future  research. 

V.  Summary  of  the  paper. 

I.   THE  EFFECT  OF  THE  INTERNAL  FRICTION  OF  FLUIDS  ON  THE 

MOTION  OF  PENDULUMS. 
I.  Observations  on  the  Motion  of  Pendulums. 

An  account  of  the  experiments  made  by  Bessel,  Baily,  Dubuat,  and 
Sabine,  and  the  theoretical  results  obtained  by  Poisson,  Challis  and 
Plana  is  given  in  Stokes's  Math,  and  Phys.  Papers,  Vol.  III.,  pp.  1-7. 

The  effect  of  the  surrounding  fluid  on  the  time  of  vibration  of  a  pendu 
lum  was  computed  by  Poisson,  Challis,  Green,  and  Plana  from  the 
hydrodynamical  theory  of  a  frictionless  fluid.     A  fair  agreement  with 
the  observations  was  found  in  some  cases,  but  in  many  cases,  especially 


493  GEORGE   F.  McEWEN.  [VOL.  XXXIII. 

where  the  dimensions  were  small,  the  theory  failed  entirely  to  account 
for  the  experimental  results. 

Because   of   this  failure,   Stokes  was  led    to  apply  the  equations   of 
motion1  of  a  viscous  fluid  to  pendulum  problems. 

2.  Stokes' s  Deduction  of  the  Law  of  Resistance.2 

In  1850  Stokes  completed  the  solution  of  the  following  problem: 
The  center  of  a  sphere  performs  small  periodic  oscillations  along  a 
straight  line;  the  sphere  having  a  motion  of  translation  only;  it  is  re- 
quired to  determine  from  the  ordinary  hydrodynamic  equations  of 
motion  of  a  viscous  fluid,  the  motion  of  the  surrounding  fluid,  and  the 
force  exerted  on  the  sphere. 

He  assumed  the  velocities  to  be  so  small  that  their  squares  could  be 
neglected,  that  there  was  no  slipping  of  the  fluid  along  the  surface  of 
the  solid  in  contact  with  it.  and  that  the  amplitude  of  vibration  of  the 
sphere  was  very  small  and  remained  constant. 
His  result  for  the  force  is 

p  A  d"y       R  dy 

F       -Adt*-Bdt> 

where  y  is  the  displacement  and  t  is  the  time,  and  the  coefficients  A  and 
B  have  the  following  values : 


where  Mi    =  the  mass  of  the  fluid  displaced  by  the  sphere, 
r  =  the  radius  of  the  sphere, 

a  =  — ,  where  T  is  the  period  of  oscillation,  X  =  ^1  — , ,  and 

fjif  =  coefficient  of  viscosity  -5-  density. 

From  the  same  assumptions  and  equations,  the  same  result  has  been 
obtained  by  a  different  method.3 

For  a  cylinder  oscillating  in  a  direction  perpendicular  to  its  axis, 
Stokes  deduced  the  following  expression : 


where  F  =  the  force  acting  upon  unit  length  of  the  cylinder,  m<l  —  the 

1  Stokes's  Math,  and  Phys.  Papers,  Vol.  I.,  pp.  75-105. 
2Stokes's  Math,  and  Phys.  Papers,  Vol.  III.,  pp.  11-36. 
8  Lamb's  Hydrodynamics,  edition  3,  pp.  583-584. 


No.  6.]  MEASUREMENTS   OF   FRICTION  A  L   FORCE.  494 

mass  of  the  fluid  displaced  by  unit  length  of  the  cylinder,  and  K«  and  KJ' 
are  constants  depending  on  the  radius  of  the  cylinder,  the  period  of 
oscillation  and  /x'. 

No  simple  expressions  for  K2'  and  K*"  were  found  that  covered  all 
cases,  and  they  are  best  determined  by  special  series  and  tables  which 
are  given  by  Stokes  in  his  Math,  and  Phys.  Papers,  Vol.  III.,  pp.  47-54. 
In  the  same  volume  he  gives  a  discussion  of  the  conditions  upon  which 
the  theory  from  which  the  above  expression  was  deduced  depends. 

In  the  case  of  the  cylinder  as  well  as  the  sphere,  the  first  coefficient 
shows  that  the  fluid  has  the  same  effect  as  an  increase  of  the  inertia  of 
the  system,  and  the  second  coefficient  shows  that  the  fluid  opposes  the 
motion  of  the  system  by  a  force  proportional  to  the  velocity. 

From  observations  on  the  time  of  vibration  the  quantity  A  can  be 
computed,  and  B  can  be  computed  from  observations  on  the  arc.  Both 
A  and  B  can  be  calculated  from  theory  with  the  aid  of  Stokes's  equations.1 
3.  The  Application  of  Stokes's  Law  to  the  Pendulum  Observations  of  Bess  el, 

Baily  and  Dubuat.2 

By  choosing  a  constant  value  of  ju'  Stokes  calculated  from  his  theory 
the  periods  of  a  variety  of  pendulums  swinging  in  air,  and  the  agreement 
with  the  values  observed  by  Bessel  and  Baily  was  very  satisfactory. 
Only  a  few  observations  on  the  arc  of  vibration  were  available,  and  these 
were  only  approximate,  in  these  cases  he  calculated  the  decrement  of 
the  arc  of  vibration  from  his  theory,  and  found  as  good  an  agreement 
with  the  experiments  as  could  be  expected  considering  the  character  of 
the  observations.  Though  most  of  the  observations  made  on  pendulums 
oscillating  in  water  were  also  in  fair  agreement  with  his  theory,  there 
were  a  number  of  discrepancies,  which  he  attributed  to  the  experimental 
methods  used. 

When  a  pendulum  oscillates  in  water,  or  a  more  viscous  liquid,  the 
arc  of  oscillation  rapidly  decreases;  this  diminution  forms,  in  fact,  the 
greatest  difficulty  in  experiments  of  this  kind.  This  difficulty,  which 
made  it  impossible  to  test  his  theory  or  to  determine  the  coefficient  of 
viscosity  of  water  or  of  more  viscous  liquids  by  means  of  a  vibrating 
body,  suggested  the  present  investigation. 

II.  THE   METHOD  ADOPTED    IN    THE    PRESENT    INVESTIGATION    FOR 

MEASURING  THE  FORCE  ACTING  ON  AN  OSCILLATING  SPHERE. 
I.  Description  of  the  Apparatus. 
A  light  rigid  bar3  FED  is  secured  to  a  knife-edge  at  £,  which  rests 

1  Stokes's  Math,  and  Phys.  Papers,  Vol.  III.,  pp.  1-141. 
« Stokes's  Math,  and  Phys.  Papers,  Vol.  III.,  pp.  76-140. 
8  See  Fig.  i  for  a  diagram  of  the  apparatus. 


495 


GEORGE   F.  McEWEN. 


[VOL.  XXXIII. 


upon  a  fixed  horizontal  support.  A  second  bar  MN,  perpendicular  to 
the  first  one,  carries  two  movable  weights  Ws  and  PF"4;  the  upper  one  can 
be  moved  to  or  from  E  by  turning  it  about  the  screw  EN,  and  a  scale 
fastened  to  MN  is  provided  for  recording  the  distance  of  Wz  from  E. 
The  lower  weight  W*  can  be  slid  along  EM  and  clamped  in  any  desired 

position.  At  Fand  D  (points  in 
line  with  JE)  two  scale-pans  W\ 
and  Wz  are  attached  by  means 
of  very  thin  strips  of  steel.  A 
second  light  rigid  bar  ABC  is 
attached  by  means  of  similar 
strips  of  steel  at  C  and  A ,  to  the 
pan  Wz,  and  to  a  support  above 
A ,  which  can  be  given  a  vertical 
periodic  motion  of  small  ampli- 
tude. This  support  does  not 
touch  the  apparatus,  except  at 
A .  The  sphere  is  suspended  in 
the  fluid  by  means  of  a  fine  wire, 
as  shown;  the  wire  being  at- 
tached by  a  thin  steel  strip,  to 
a  small  support  B,  which  can  be 
fastened  to  the  bar  AC,  in  any 
position  between  A  and  C,  but 
is  always  in  line  with  A  and  C. 

By  adding  weights  to  the 
scale-pans,  the  system  can  be 
brought  to  a  state  of  equilibrium 
when  FD  is  horizontal.  By  ad- 
justing the  weights  W*  and  W^ 
the  natural  period  of  oscillation 
can  be  varied,  as  this  adjustment  changes  the  height  of  the  center  of 
gravity  of  the  system  rigidly  attached  to  the  knife-edge. 

The  oscillations  will,  from  the  construction  of  the  apparatus,  cause 
the  sphere  to  move  in  a  vertical  line,  and  if  the  support  of  A  is  fixed, 
these  oscillations  will  subside  because  of  the  friction.  But  if  the  support 
of  A  is  given  a  vertical  periodic  motion,  the  system  will  be  set  in  forced 
vibration,  and  the  amplitude,  phase  and  period  of  this  forced  vibration 
will  depend  upon  the  amplitude,  phase  and  period  of  the  motion  of  the 
support,  for  any  given  adjustment  of  the  vibrating  system. 


*—  h 


Fig.  i. 


No.  6.]  MEASUREMENTS   OF   FRICTIONAL   FORCE.  496 

2.  The  Theory  of  the  Forced  Vibrations  of  the  Above  System. 

Explanation  of  the  Symbols: 

M"  equals  the  mass  rigidly  connected  to  the  knife-edge  E. 

Ic"  equals  the  moment  of  inertia  of  M"  about  C\. 

IE"  equals  the  moment  of  inertia  of  M"  about  E. 

Ci  is  the  center  of  mass  of  M"  '. 

ECi  equals  /. 

ms'  equals  the  mass  of  the  lower  bar. 

73'  equals  the  moment  of  inertia  of  m3f  about  its  center  of  mass. 

3>3  equals  the  displacement  of  the  center  of  mass  of  m-l  from  the  position 
of  equilibrium. 

Mi  equals  the  mass  of  the  sphere  and  its  suspending  wire. 

6  equals  the  angular  displacement  of  FD  from  its  equilibrium  position 
which  is  assumed  to  be  horizontal. 

Mi  equals  the  mass  of  the  fluid  displaced  by  the  sphere. 

g  equals  the  acceleration  of  gravity 

2 

5  =  s»  +  s3;  h  =  --  s2. 

do  equals  the  angular  displacement  of  the  heavy  pendulum  used  to 
maintain  the  vibrations. 

—  /i00  equals  yo,  the  linear  displacement  of  the  point  A. 

|8  equals  the  change  in  torque  due  to  the  elasticity  of  one  suspending 
strip  when  bent  through  unit  angle. 

&\y  equals  the  change  in  the  force  on  the  wire  holding  the  sphere,  due 
to  the  elasticity  of  the  surface  film  of  the  liquid,  and  to  the  varying  length 
of  the  wire  immersed. 

The  variable  force  exerted  on  the  sphere  by  the  fluid  equals 


Denote  the  variable  force  exerted  on  the  wire  by  the  fluid  by: 


<£  equals  the  angular  displacement  of  the  lower  bar  from  the  equilibrium 
position. 

d<f> 
—  kz  ~r  =  the  torque  on  the  lower  bar  due  to  the  friction  of  the  air 

and  the  suspending  wires. 

—  ki  -,-  =  the  torque  on  the  upper  bar  due  to  the  friction  of  the  air, 
the  suspending  wires,  and  the  knife-edge. 


497  GEORGE   F.  McEWEN.  [VOL.  XXXIII. 

Fg  equals  the  horizontal  force  at  E  acting  on  the  knife-edge. 
FK  equals  the  vertical  force  at  E  acting  on  the  knife-edge. 
FA  equals  the  vertical  force  at  A  acting  on  the  lower  bar. 

—  FB  equals  the  vertical  force  at  B  acting  on  the  lower  bar. 
Fc  equals  the  vertical  force  at  C  acting  on  the  lower  bar. 

—  FD  equals  the  vertical  force  at  D  acting  on  the  upper  bar. 

—  F  equals  the  vertical  force  at  F  acting  on  the  upper  p«*£. 

ms  equals  the  mass  of  each  scale-pan  and  contents,  and  wires  between 
C  and  D. 

m±  equals  the  mass  of  each  scale  pan  and  contents,  and  wires  attached 
at  F. 

The  meaning  of  the  other  symbols  used  is  indicated  on  the  diagram 
of  the  apparatus. 

Fundamental  Dynamical  Relations: 


Mi'g  -(A+A')2-(B  +  B') 

-B 

~     =  -  FB  +  FA  +  Fc  -  m,fgt 


Io"^  =  -  Fj>Si  +  FS*  -  FE'L  cos  6  -  FEL  sin  6  -  k^  -  2/30 
+  FL  sin  6  +  /y  sin  0, 

M"  ^  =  K"  I  (cos  0)  (^  )V  (sin  0)  ^  J  /  =  FE  -  F  -  ^  -  M"g, 


c  ,  „   .      -  ^ .  ,      ,  de 

—  01 1  rc 


{?2>3 
(cos0)^-(sin 

f        r          /  do  \ 2  d*d  i  1 

/(sin  e  I  M"l  [  (cos  0)  (  d-  )   +  (sin  0)  ^  -  J  +  M"g  | 


No.  6.]  MEASUREMENTS   OF   FRICTIONAL  FORCE.  498 

Fundamental  Kinemalical  Relations: 
x  and  y  are  the  coordinates  of  C\, 

x  =  I  sin  6,  y  —  —  I  cos  9, 

dx  d9       dy  ^0 

dt=l(c°s6}dt<     dt   =  l(sm^dt' 


d$ 


ofl      dy*_        o^       <Py< 
"    -S'9'      dt  -     ~Stdt'      df 


y  = 


9  <?  7 

slS2di*~S3llW 


y»  _  if  ~<_,  <h\ 
d?  "~  2\   '*2  "''  dp)' 

d<i> 

dt  ~ 


d?  '  6l 

The  above  dynamical  and  kinematical  relations  can  be  so  combined 
as  to  give  the  following  ordinary  differential  equation  : 


499  GEORGE   F.  McEWEN.  [VOL.  XXXIII. 

in  which  the  coefficients  are  constants  and  have  the  following  values: 
P  =  [/*"  +  m*S?  +  w3Si2  +  Sl^  (M,  +  A  +  A')  +  S~m,f  +  ^'~], 


Assume 

jo  =  ce~a'  sin  at  =  —  lido,     • 

and  substitute  in  the  second  member  of  the  differential  equation;  the 
result  will  be  : 


c°s 

where 


P0"  =  (P2  _  [as  _  o2]p"  _  ap')f     and     P0'  =  o(P'  - 

Assume  that  the  following  ordinary  equation  is  a  particular  integral 
of  the  differential  equation  : 

6  =  Atf~at  sin  (at  -  61)  +  B,e~at  cos  (at  -  02), 
substitute  in  the  equation  and  solve  for  the  quantities: 

Ai,  0i,  jBi,  and  02- 

The  following  values  of  the  constants  will  make  the  assumed  form  for 
0,  a  solution: 


P2  +  P[«2  -  a2]  -  «Pi  +  (P\a  -  2aaP)  tan  0t ' 


P2  +  P[a2  -  a2]  -  aPi  +  (Pia  -  2ao:P)  tan  02 ' 
tan  0i  =  - 


No.  6.]  MEASUREMENTS   OF   FRICTION  A  L   FORCE.  5OO 

To  obtain  the  complete  integral  of  the  equation,  the  complementary 
function,  Aae~Plt/zp  sin  (&/  +  03)  would  have  to  be  added  to  the  above 
value  of  0.  A$  and  03  are  arbitrary  constants,  depending  for  their  values 
on  how  the  system  is  started,  and 


= 


2P 


In  the  present  method  of  measurement,  no  observations  are  taken 
until  the  complementary  function  has  a  negligible  value,  so  only  the 
first  or  particular  integral  will  be  retained. 

In  practical  applications,  the  ratio  of  the  maximum  value  of  0  to  the 
maximum  value  of  0o  is  determined  by  experiment.  The  relation  of  the 
coefficient  B  to  that  ratio,  which  is  required  for  determining  B  will  now 
be  derived.  The  following  equation  for  0o  is  assumed: 

0o  =  —  r  e~at  sin  at. 
l\ 

Then  the  expression  for  0  can  be  reduced  to  the  form 

0  =  A2e~at  sin  (at  -  0i  +  03) 
where 


+  B  i2,     and     tan  03  =  - 


Al      P»"' 

Let  t\  be  the  time  at  which  0  has  a  maximum  value,  and  k  be  the  time 
at  which  0o  first  reaches  its  maximum  value  after  t\.  Let  k  —  ti  =  A/. 

By  taking  the  first  derivative  of  0  and  0o,  the  maximum  values,  0' 
and  0o'  given  below  are  obtained : 

0'  =  A2e 


Dividing  the  first  equation  by  the  second  gives 


- 

9J 


This  ratio  is  obtained  experimentally  by  first  observing  the  maximum 
value  of  0,  then  observing  the  next  maximum  value  of  0o,  and  dividing 
the  first  by  the  second.  Substituting  the  value  of  A  and  B  in  the  expres- 
sion for  A  gives 

2  +  (A")2 


a2] 


5<DI  .    GEORGE   F.  McEWEN.  [VOL.  XXXIII. 

and 

v/(P2  +  P[<*2  -  a2]  -  aPl)2~+~(JPla^2aaP^ ' 

This  expression  shows  that  R'  is  independent  of  the  amplitude  of  the 
vibration,  but  is  a  function  of  the  constant  coefficients  of  one  of  the 
preceding  differential  equations. 

3.  The  Deduction  of  a  Practical  Working  Formula  from  the  Above  Theory. 
It  appears  that  P2  is  the  constant  by  which  6  is  multiplied  to  obtain 
that  part  of  the  torque  which  opposes  the  displacement  of  the  system, 
but  does  not  depend  on  the  friction.     In  the  expression 


/  is  the  distance  above  E  of  the  center  of  mass  of  the  system  rigidly 
attached  to  the  knife-edge.  As  will  be  seen  by  referring  to  the  expression 
for  Rf,  for  the  values  of  the  constants  in  the  expression  for  R',  the  quan- 
tity /  appears  only  in  the  expression  for  P2.  Therefore,  by  shifting  the 
weight  Wz,  I  and  consequently  M"gl  can  be  changed  without  affecting  the 
remainder  of  the  expression  for  R'.1 

Let 


*-<— '#..  _,2  +  (Po")2 


f  P[a2  -  a2]  -  aPO2  H 

now  if  R  is  measured  for  three  different  values  of  M"gl,  there  will  be 
three  equations  from  which  PI  can  be  computed  by  eliminating  the 
other  unknown  quantities. 

Suppose  M"gl  is  changed  by  raising  or  lowering  the  weight  WB,  a 
known  amount;  the  amount  due  to  one  complete  turn,  for  example. 
Denote  this  change  by  */&.  Then 

where  /  is  the  distance  of  the  center  of  the  mass  of  M"  from  E,  /  is  the 
distance  of  the  center  of  mass  of  (M"  —  W$)  from  £,  and  k  is  the  distance 
of  the  center  of  mass  of  W*  from  E.  Let  the  change  in  /3  be  A/3,  and 
denote  the  corresponding  change  in  /  by  A/,  then  /  will  remain  the  same 
as  it  depends  only  on  that  part  of  the  mass  whose  position  with  reference 

1  Changing  I  in  this  way  would  produce  a  known  small  change  in  P,  which  can  be  easily 
compensated  for  by  changing  the  weights  in  the  scale-pans  Wi  and  Wz.  It  will  be  assumed 
that  this  is  done,  and  therefore  P  will  be  treated  as  a  constant. 


No.  6.]  MEASUREMENTS   OF   FRICTION  AL   FORCE.  502 

to  E  is  fixed.     Therefore, 

M"(l  +  A/)  =  (M"  -  Wi)l  +  Wz(h  +  A/3), 


and 


In    the   equation    for   R,    denote    the    numerator    by    v/23>   denote 
(aPi-2a«P)2  by  zi,  denote  (P2  +  P[«2  -  a2]  -  aPi)  by  H^ZZ  -  z',1  and  let 

i  —  -  =  Wm     Then 

»  A     /    ..  . 


(P2  +  P[<*2  -  fl2]  -  c 
and  the  equation  for  R  takes  the  following  simplified  form : 

2  z8 

Now  by  varying  n,  the  number  of  complete  turns  of  Ws,  Rn  will  vary 
but  22,  z3,  and  z\  will  remain  constant,  and  are  positive,  w  is  a  con- 
stant, positive  or  negative,  and  depends  on  the  position  of  PF3  when  n  is 
assumed  to  be  zero. 

The  maximum  value  of  R,  denoted  by  Rw  is : 

Rw  = 

therefore  Rw  is  a  function  of  z\  and  z3  only,  and  corresponds  to  the  special 
case  when  there  is  resonance.  The  above  equation  involving  Rn  can  be 
written  in  the  following  forms: 


(n  -  w)  =  WF  -  I  =  ± 


-71 


(n  - 


where  V  =  —  . 

22 

An  inspection  of  the  above  equations  shows  that  if  Rn  and  Rw  are  both 
multiplied  by  an  arbitrary  constant,  the  roots  of  the  equation  will  be 

1  Assume  that  A/j  corresponds  to  one  turn  of  Ws,  and  denote  by  n  the  number  of  turns 
from  a  given  position  near  the  top  of  the  screw.     Then  n  and  P»  increase  as  Wi  is  lowered. 


503  GEORGE   F.  McEWEN.  [VOL.  XXXIII. 

unaltered.  Therefore  the  ratios  used  need  only  be  proportional  to  the 
true  values  of  the  ratios.  Also,  if  (A)  has  a  constant  amplitude,  only 
the  readings  for  6  need  to  be  taken. 

From  three  observations,  three  correspondeing  values  of  n  and  Rn 
can  be  obtained,  and  the  two  quantities  Rw  and  w,  which  will  be  the  same 
in  all  cases  can  be  eliminated,  thus  giving  the  value  of  ^  V. 

Now, 

V^  =  y/~V<S~zz  =  aPi  -  2aaP, 
and  therefore 


also 


and 


where  K  is  a  positive  constant,  and  depends  only  on  s,  $1,  s*,  a,  P  and 
the  friction  not  due  to  the  sphere.  Therefore,  if  two  different  spheres 
of  radii  r\  and  rz  are  used, 


and 


where  Ki  —  K\  is  practically  zero.     Therefore,  subtracting  the  first  from 
the  second  gives 


From  this  equation,  the  difference  between  52  and  BI  can  be  found, 
since  all  the  quantities  in  the  second  member  can  be  measured.  From 
the  theory  given  by  Stokes,  this  difference  has  the  following  value  : 


-  n)  ( 


ft  -I- 


and  from  this  equation  /x  the  coefficient  of  viscosity  can  be  computed. 
But  the  vessel  containing  the  liquid  must  be  large  enough,  compared 
to  the  sphere  used  so  that  the  assumption  on  which  the  equations  were 


No.  6.[  MEASUREMENTS   OF   FRICTION  A  L  FORCE.  504 

derived  are  justified,  or  a  correction  for  the  effect  of  the  containing  vessel 
must  be  made. 

Let  Bi  and  B2'  be  the  values  of  BI  and  Bz  corresponding  to  a  vessel 
of  infinite  radius.  Let  Xi  and  X2  be  coefficients  depending  on  a,  \L  ,  and 
the  ratio  of  the  radius  of  the  sphere  to  the  containing  vessel,  so  that 

\iBi   =  BI, 
and 

\2-O2      =    B%, 

then 


Let  Xi"  and  X2/;  correspond  to  the  cases  where  the  radius  of  the  vessel  is 
R"  and  Rf  respectively;  the  same  sphere  being  used.     Then 


If  this  result  is  zero,  then  the  radius  is  large  enough,  and  no  correction 
is  necessary. 

III.   EXPERIMENTAL  RESULTS. 

A  gravity  pendulum  consisting  of  a  wrought  iron  rod  1.5  cm.  in  diam- 
eter, and  1  80  cm.  in  length;  and  two  cylindrical  weights,  each  being  of 
5  kg.  mass,  was  mounted  on  a  frame  supported  by  two  concrete  piers. 
By  adjusting  the  position  of  the  weights,  any  value  of  the  period,  from 
about  2  to  25  seconds  could  be  secured.  By  means  of  a  mirror  attached 
to  the  knife-edge,  and  a  vertical  scale  and  horizontal  telescope  supported 
on  one  of  the  piers,  115  cm.  from  the  mirror,  measurements  of  the  angular 
displacements  were  made.  The  scale  was  divided  into  mm.,  and  readings 
were  taken  to  o.i  mm.  Within  the  limits  of  observational  errors,  the 
displacement  of  the  pendulum  agreed  with  the  following  formula: 


*=(-;,) 


r  I  e  at  sin  at. 


The  measuring  apparatus,  already  described,  was  supported  with  its 
knife-edge  E  parallel  to  that  of  the  large  pendulum,1  and  about  8  cm. 
above  it.  A  mirror  was  attached  to  E  above  the  mirror  of  the  large 
pendulum,  and  a  second  horizontal  telescope  was  mounted  above  that 
used  for  the  large  pendulum.  The  same  scale  was  used  for  both.  The 
apparatus  was  not  enclosed  nor  shielded  from  air  currents. 

A  short  rigid  bar,  secured  to  the  large  pendulum  in  a  position  perpen- 
dicular to  its  knife-edge  and  pendulum  rod,  supported  at  the  distance  /i 

1  See  Fig.  2  for  diagram. 


505 


GEORGE  F.  McEWEN. 


[VOL.  XXXIII. 


from  the  knife-edge,  by  means  of  a  fine  wire,  the  end  A  of  the  lower  beam 
of  the  measuring  apparatus.  Because  of  this  arrangement,  the  end  A 
of  the  beam  AC  was  constrained  to  have  the  vertical  displacement: 

yo  =  —  /i0o  =  ce~at  sin  at. 

In  order  to  test  the  capacity  of  the  apparatus  for  measuring  large 
forces  acting  on  the  sphere,  dark  filtered  cylinder  oil,  grade  "N,"  was 
used  at  temperatures  from  18°  to  22°  Cent.  At  these  temperatures, 
the  coefficient  of  viscosity  of  the  oil  is  several  thousand  times  as  great 
as  that  of  water. 

The  numerical  values  of  the  constants  in  the  experiment  on  oil  are 
tabulated  below: 

Weight  of  the  measuring  apparatus,  about  400  grams. 


Weight  in  each  scale-pan,  about 
Weight  of  each  scale-pan,  about 

W3  =  13.52  gr. 

A/3  =  00.0794  cm- 
Si  =  5.0  cm. 
53  =  7.5  cm, 
a  =  1.702  (sec.)"1. 


80  grams. 
15  grams. 

M"  =  76.84  gr. 

^z2  =  1,052  dyne  cm, 

s2  =  2.5  cm. 

5  =  10.0  cm. 

a  =  0.0007    (sec.)"1, 


The  radius  of  the  cylinder  containing  the 
liquid  was  5.5  cm.;  its  depth  was  18  cm. 
The  cylinder  was  practically  full  of  the  liquid, 
and  in  all  cases  the  top  of  the  sphere  was  6.5 
cm.  below  the  liquid  surface.  The  maximum 
velocity  of  the  sphere  was  less  than  o.i  cm./ 
sec.,  and  itsmaximum  displacement  was  less 
than  0.07  cm. 

The  maximum  deflections  6'  and  00'  were 
determined  alternately,  and  the  value  for  0' 
was  divided  by  the  mean  of  the  two  nearest 
values  of  0o'  for  a  single  determination  of  the 
ratio  R'.  The  following  set  of  readings  and 
results  is  typical  of  the  accuracy  of  the  work.  In  the  first  and  second 
columns  are  the  average  of  the  differences  of  the  scale  readings  for 
determining  00'  and  00  respectively.  The  last  column  contains  the  average 
value  of  Rf  minus  each  value. 


P.     2 


No.  6.1 


MEASUREMENTS   OF   FRICTIONAL   FORCE. 


506 


R' 

.569*-*' 

R' 

.569*-*' 

2.590 
2.575 
2.560 
2.540 
2.515 
2.480 

1.560 
1.520 
1.510 
1.505 
1.495 
1.485 

.6025 
.5910 
.5900 
.5925 
.5945 
.5990 

-.0063 
.0048 
.0062 
.0037 
.0017 
-.0028 

2.445 
2.425 
2.400 
2.370 

1.460 
1.460 
1.430 
1.420 

.5970 
.6062 
.5960 
.5980 

-.0008 
-.0058 
.0002 
-.0018 

.5962 

Tabulation  of  Results. — The  temperature  of  the  liquid  is  denoted  by  /. 
Under  V  V  are  the  results  of  substituting  the  given  values  of  w,  w,  Rn, 
and  Rw  in  the  formula: 

*<»-•> 


/  =  18.1°       n  =  .495  cm.     RJ    =  8.30     w  =  1.384. 


n 

*n 

*n» 

1*\ 

£i+A"=39S.7^i 

1 

2.612 

6.822 

2.973 

7 

1.5513 

2.4065 

2.950 

11 

.9375 

.8789 

2.965 

17 

.5709 

.3259 

2.954 

2.960 

1,171 

=  18.05°     r2  =  .792  cm.     Rw*  =  10.12     w  =  -1.700. 


n 

*« 

*n« 

<v* 

*+«-•**: 

0 

3.123 

9.755 

8.79 

6 

2.392 

5.722 

8.78 

18 

1.296 

1.680 

8.79 

I        8.787 

3,477 

22.3 


.495  cm.     RJ  =  11.30     w  =  2.98. 


• 

*» 

£ 

VFi 

*M 

1 

1.9817 

3.927 

1.445 

7 

1.1335 

1.2848 

1.436 

11 

.5792 

.33507 

1.405 

17 

.3423 

.11717 

1.436 

1.431 

566 

22.15°     n  =  .792  cm.     l?w»  =  9.34     w  -  1.50. 


« 

/?« 

*«« 

g 

WMT 

0 

2.755 

7.590 

3.125 

6 

1.748 

3.055 

3.137 

12 

.9001 

.8102 

3.237 

18 

.5701 

.3255 

3.136 

1 

3.159 

1,249 

507 


GEORGE   F.  McEWEN. 


[VOL.  XXXIII 


t  =  22.10° 


.792  cm.     /?,,2    =  8.63     w  =  1.73. 


n 

^n 

Rn* 

V  ^ 

B^K 

1 

2.862 

8.191 

3.155 

7 

1.513 

2.289 

3.142 

11 

.9500 

.9025 

3.160 

17 

.5962 

.3555 

3.162 

1           i    3.155 

1,248 

The  coefficient  of  viscosity  was  not  computed  from  the  preceding 
data,  because  the  cylinder  was  not  large  enough  compared  to  the  spheres 
for  the  assumption  of  a  cylinder  of  infinite  radius  to  be  valid  for  a  fluid 
having  such  a  large  coefficient  of  viscosity.  However,  the  above  trials 
show  that  the  method  adopted  can  be  successfully  applied,  even  to  the 
measurement  of  very  large  frictional  forces. 

The  apparatus,  used  in  the  experiment  just  described,  was  not  well 
adapted  to  the  measurement  of  small  forces  acting  on  the  sphere,  because 
of  its  crudeness,  and  because  the  friction  in  the  apparatus  itself  was  com- 
paratively large.  But  in  order  to  test  the  method  through  a  wide  range 
of  conditions,  an  attempt  was  made  to  determine  the  coefficient  of 
viscosity  of  water.  In  this  experiment  the  sphere  was  suspended  close 
to  the  point  C  of  the  lower  bar,  in  order  to  increase  the  effect  of  the  fric- 
tional force.  Also,  since  the  quantity  z\  was  now  much  smaller  than 
before,  the  series  of  values  of  R  was  obtained  by  correspondingly  smaller 
changes  in  the  expression  (P2  +  P[a2  —  a2]  —  aPi).  The  changes  were 
made  by  adding  equal  weights  to  each  scale-pan,  thus,  varying  P  instead 
of  Pz  as  was  done  before;  and  by  a  method  similar  to  that  already 
described  the  following  formula  was  derived : 


in  which  n  is  the  number  of  multiples  of  a  given  weight  W±  added  to 
each  scale-pan,  v/22  equals  the  change  in  the  moment  of  inertia  of  the 
system  when  one  of  these  weights  is  added  to  each  pan.  That  is: 


In  this  case  a  decrease  in  n  has  the  same  effect  as  an  increase  in  the 
previous  case,  but  this  formula  can  be  used  in  the  same  way  as  the 
previous  one,  assuming  the  symbols  to  have  the  values  indicated  above. 

The  numerical  values  of  the  constants  in  the  following  experiment 
are  tabulated  below: 


Weight  of  the  measuring  apparatus,  about  400  grams. 


No.  6.] 


MEASUREMENTS   OF   FRICTIONAL   FORCE. 


508 


Weight  in  each  scale-pan,  about  50  grams. 

Weight  of  each  scale-pan,  about  15  grams. 

W\  —  i.oo  grams.  ^z2  =  146.5  dyne  cm. 


Si  =  5.0  cm. 


=  I.I  cm. 


p  =  i.ooo. 


=  8.90  cm. 


=  10.0  cm. 


a  =  1.711  (sec.)"1. 


The  radius  of  the  cylinder  containing  the  liquid  was  8  cm.;  its  depth 
was  20  cm.  The  cylinder  was  practically  full  of  water,  and  in  each  case 
the  top  of  the  sphere  was  6.5  cm.  below  the  surface  of  the  water.  The 
maximum  velocity  of  the  sphere  was  less  than  .15  cm./sec.,  and  its 
maximum  displacement  was  less  than  .09  cm. 

The  amplitude  of  the  large  pendulum  was  maintained  constant  by 
means  of  an  electro-magnet  and  the  scale  readings  for  determining  6 
were  taken  as  before.  In  this  case  they  should  be  constant.  The  fol- 
lowing set  of  readings  and  results  is  typical  of  the  accuracy  of  the  work. 
The  differences  between  successive  scale  readings,  denoted  by  R,  are  given 
in  the  first  column.  The  last  column  contains  the  average  value  of  R 
minus  each  value. 


R 

2.719-;? 

R 

2.719-;? 

R 

2.719—;? 

2.67 

+  .049 

2.74 

-.021 

2.68 

+  .039 

2.70 

+  .019 

2.73 

-.011 

2.67 

+  .049 

2.68 

+  .039 

2.68 

+  .039 

2.68 

+.039 

2.69 

+  .029 

2.69 

+  .029 

2.71 

+  .009 

2.68 

+.039 

2.72 

-.001 

2.76 

-.041 

2.72 

-.001 

2.73 

—  .011 

2.76 

-.041 

2.79 

-.071 

2.69 

+.029 

2.78 

-.061 

2.77 

-.051 

2.72 

—  .001 

2.71 

+.009 

2.78 

-.061 

2.71 

+.009 

2.75 

-.031 

2.77 

-.051 

2.71 

+  .009 

2.70 

+  .019 

2.72 

-.001 

2.70 

+.019 

Aver.  R  2.719 

2.72 

-.001 

TABULATION  OF  RESULTS. 

The  temperature  of  the  liquid  is  denoted  by  /.     Under  ^  V  are  given 
the  results  of  substituting  the  values  of  n,  w,  Rnt  and  Rw  in  the  formula: 


"  -  R 


509 


GEORGE   F.  McEWEN. 
t  =  16.5°     r  =  .495  cm.     RJ    =  85.8     w  =  3.474. 


[VOL.  XXXIII. 


n 

*. 

JW 

VVi 

*+**MI^ 

4 

6.994 

48.95 

.6057 

2 

3.525 

12.42 

.6065 

0 

1.593 

2.538 

.6065 

.606 

1.530 

t  =  16.5°     r2  =  .952  cm.     Rtv*    =  42.80     w  =  4.34. 

« 

*„ 

Jtf 

V  V* 

*+A-=*S*/* 

8 

1.850 

3.422 

1.079 

4 

6.238 

38.92 

1.080 

2 

2.721 

7.405 

1.070 

0 

1.556 

2.420 

1.064 

1.073 

2.710 

B2  —  Bi  =  2.710  —  1.530  =  1.18. 
The  expression  (J?2  —  -Si)  can  be  transformed  into  the  following 


-  Bl  = 


-  n) 


from  which  after  substituting  the  values  from  the  above  experiment,  the 
following  equation  for  determining  ju  is  obtained  : 


or 


1.18  =  STT  ^3.422  \V{ (457) (1447  +  1. 
M'  -f  I-338  ^M7  -  -1369  =  o. 


The  solution  of  this  equation  gives  ju  =  .0091,  while  the  value  .01 1 1  is 
given  by  the  formula:1 

.0178 

~  I   +  .0337/  +  .00022 It2' 

Thus  there  is  a  fair  agreement  with  other  methods,  considering  the  un- 
favorable working  conditions. 

IV.  SUGGESTIONS  FOR  FUTURE  RESEARCH.2 

In  the  measuring  apparatus  used  in  the  experiments  just  described, 
the  force  which  opposed  the  displacement  of  the  system  but  did  not 

1  This  formula  is  based  on  Poiseuilles'  capillary  tube  experiment.  See  Lamb's  Hydro- 
dynamics, 3d  edition,  p.  536. 

8  Further  suggestions  for  future  research  are  given  in  Stokes's  Math,  and  Phys.  Papers, 
Vol.  III.,  pp.  123-127. 


No.  9.1  MEASUREMENTS   OF    FRICTION AL   FORCE.  510 

depend  upon  the  friction,  was  due  partly  to  the  elasticity  of  the  steel 
strips  and  partly  to  gravity.  If,  instead  of  steel  strips  knife-edges  were 
used,  the  return  force  would  depend  upon  gravity  only  and  could  be 
made  so  small,  without  loss  of  stability,  that  a  given  natural  period  could 
be  secured  by  a  much  lighter  apparatus.  If,  in  addition  to  reducing  the 
weight,  agate  bearings  were  provided,  the  friction  of  the  apparatus  would 
be  greatly  reduced,  and  therefore  greater  accuracy  would  result.  It 
would  also  be  desirable  to  protect  the  apparatus  from  air  currents  and 
dust. 

In  making  measurements,  the  weight  W$  should  be  so  adjusted  that 
two  values  of  the  ratio  R  are  roughly  one  half  the  maximum  value,  one 
corresponding  to  the  case  where  (P2  +  P[a2  —  #2]  —  aPi)  =  v' ' Zi(n  —  w)  is 
positive  and  the  other  when  it  is  negative,  and  the  third  value  should  be 
near  the  maximum.  The  three  values  of  R  thus  obtained  are  sufficient 
for  computing  the  force,  but  it  is  desirable  to  check  the  work  by  addi- 
tional values  of  R  between  the  first  two. 

The  same  apparatus  can  be  used  for  the  measurement  of  the  force 
acting  on  any  vibrating  body,  if  the  force  opposes  the  displacement  and 
is  proportional  to  the  velocity,  and  it  is  well  adapted  to  the  case  in 
which  a  very  small  maximum  velocity  and  displacement  is  required. 

The  advantages  of  measuring  the  coefficient  of  viscosity  by  the  oscil- 
lating sphere  method  can  only  be  settled  by  a  series  of  experiments  in 
which  the  force  is  measured  when  the  working  conditions  are  in  agree- 
ment with  those  required  by  the  theory. 

V.   SUMMARY  OF  THE  PAPER. 

The  hydrodynamical  theory  of  a  frictionless  fluid  failed  to  account 
for  the  experimental  results  of  the  pendulum  observations  of  Bessel, 
Baily  and  Dubuat,  and  this  fact  led  Stokes  to  apply  the  theory  when  the 
fluid  friction  was  taken  into  account. 

The  predictions  based  on  this  theory  agreed  well  with  the  experiments 
on  pendulums  vibrating  in  gases,  but  when  the  fluid  was  water,  instead 
of  gas,  the  agreement  was  not  very  satisfactory.  This  disagreement 
was  due  to  the  difficulty  of  making  the  observations,  rather  than  to  the 
theory. 

The  fact  that  the  inertia  correction  could  be  determined  experimentally 
and  that  it  had  been  shown  how  it  depended  on  the  coefficient  of  vis- 
cosity led  Stokes  to  suggest  that  his  theory  might  be  employed  to  cal- 
culate the  coefficient  of  viscosity  of  gases  from  observations  on  a  pendu- 
lum, consisting  either  of  a  sphere  attached  to  a  fine  wire,  or  of  a  cylinder. 

He  also  showed  how  to  compute  the  frictional  force  acting  on  a  sphere 


511  GEORGE   F.  McEWEN.  [VOL.  XXXI  II 

and  a  cylinder,  vibrating  in  a  viscous  fluid.  From  this  result  the  coef- 
ficient of  viscosity  can  be  computed  from  observations  which  give  the 
retarding  force.  For  this  purpose  he  proposed  that  the  decrement  of 
the  arc  of  oscillation  be  used.  This  does  not  require  as  accurate  a  value 
of  the  period  as  the  other  method. 

The  greatest  difficulty  in  each  method  when  the  coefficient  of  viscosity 
is  as  large  as  that  of  water,  or  larger,  is  the  rapidity  with  which  the 
oscillations  diminish. 

The  present  investigation  was  undertaken  to  overcome  the  difficulties 
mentioned  by  Stokes  when  applying  the  oscillating  sphere  method  to  the 
measurement  of  large  coefficients  of  viscosity.  This  object  was  accom- 
plished by  employing  a  forced  vibration  method  in  which  the  sphere  was 
suspended  by  a  fine  wire,  and  had  a  slow  motion  of  translation  only 
along  a  vertical  line,  and  the  oscillations  were  small,  and  either  diminished 
very  slowly,  or  were  maintained  constant  for  any  desired  length  of  time. 

The  above  method  was  applied  to  the  measurement  of  the  force  acting 
on  a  sphere  in  water  and  in  a  very  viscous  oil.  In  each  case  a  satisfactory 
measurement  of  the  force  was  obtained. 

It  is  assumed  in  the  hydrodynamical  theory  that  the  velocities  and 
displacements  are  very  small,  that  the  amplitude  remains  constant,  and 
that  there  is  no  rotation  of  the  sphere.  In  measuring  the  coefficient  of 
viscosity  it  is  necessary  that  the  apparatus  can  be  easily  cleaned,  that 
the  required  quantities  can  be  accurately  measured,  and  that  the  condi- 
tions can  be  readily  reproduced. 

These  requirements  are  fulfilled  by  the  method  just  described,  but 
the  advantages  of  measuring  the  coefficient  of  viscosity  by  the  oscillating 
sphere  method  can  only  be  determined  by  further  experiments  in  which 
the  force  is  measured  when  the  conditions  agree  with  those  required  by 
the  theory. 


9 

University, 
rtl  ,  11/1. 


THE   PHYSICAL   REVIEW 

A  Journal  of  Experimental  and  Theoretical  Physics 
•     Conducted  with  the  Co-operation  of  the 

AMERICAN  PHYSICAL  SOCIKTY 

By  EDWARD  L.  NICHOLS 
ERNEST  MERRITT,  AND  FREDERICK  BEDELL 

Advisory  Editors  for  Current  Volume  : 

J.  S.  AMES  W.  F.  MAGIE  B.  O.  PEIRCE 

K.  E.  GUTHE  R.  A.  MILLIKAN  C.  A.  SKINNER 

J.  C.  MCLENNAN  E.  F.  NICHOLS  J.  ZELENY 


IN  THE  UNITED  STATES 
Annual  Subscription  $6.00  Single  Copies,  $  .60 

OUTSIDE  THE  UNITED  STATES 
Annual  Subscription  $6.50  Single  Copies,  $  .65 

Two  volumes  of  THE  PHYSICAL  RBVIBW  are  published  annually,  these  vol- 
umes beginning  in  January  and  July,  respectively,  and  containing  six  numbers 
each.  Subscriptions  should  be  sent  to  the  publishers,  THE  PHYSICAL  RE- 
VIEW, 41  North  Queen  Street,  Lancaster,  Pa.,  or  Ithaca,  N.  Y.  /  or  to  Messrs. 
MAYER  AND  MUELLER,  Berlin.  In  order  that  subscriptions  may  terminate 
with  the  December  issue,  subscriptions  for  less  than  one  year,  terminating  in 
December,  will  be  taken  at  the  yearly  rate. 

Previous  to  Volume  V  (July-December,  1897)  THE  PHYSICAL  REVIEW  was 
published  in  annual  volumes,  each  containing  six  bi-monthly  numbers,  begin- 
ning with  the  July-August  number,  1893. 

A  limited  number  of  complete  sets  of  the  REVIEW  will  be  supplied  at  the 
price  of  $3.00  per  volume  for  all  volumes,  carriage  extra.  When  they  can  be 
furnished  without  breaking  a  set,  separate  back  volumes  will  be  supplied  at 
the  same  price,  and  single  back  numbers  will  be  mailed  to  any  address  in  the 
United  States  at  the  price  of  $.60  per  copy,  or  to  any  address  outside  the 
United  States  at  the  price  of  $.65  per  copy. 

For  advertising  rates,  address  THE  PHYSICAL  REVIEW,  Ithaca,  N.  Y. 
Change  of  copy  in  standing  advertisements  should  be  sent  direct  to  the  print- 
ers, THE  NEW  ERA  PRINTING  Co.,  Lancaster,  Pa. 

Correspondence  relating  to  contributions  should  be  addressed  to  the  editors 
at  Ithaca,  New  York.  If  an  article  submitted  for  publication  in  the  REVIEW 
is  to  be  published  elsewhere,  notice  to  that  effect  should  be  given.  All  manu- 
script submitted  should  be  ready  for  the  printer;  the  editors  cannot  assume 
responsibility  for  its  correctness.  Illustrations  should  be  in  black  and  white, 
and  ready  for  reproduction.  Curves  should  be  plotted  on  plain  paper  with 
ruled  black  co-ordinates  or  on  blue-lined  cross-section  paper.  Cross-section 
paper  with  lines  of  any  other  color  should  be  avoided.  Offprints,  when  or- 
dered in  advance,  may  be  obtained  at  prices  depending  upon  die  length  of  the 
article,  etc.  A  circular  containing  prices  of  offprints  and  other  information  for 
contributors  may  be  obtained  from  the  editors. 


Entered  at  the  pott-office,  at  Lancaster,  Pa.,  as  second  class  nutter. 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


AN  INITIAL  FINE  OF  25  CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
WILL  INCREASE  TO  SO  CENTS  ON  THE  FOURTH 
DAY  AND  TO  $1.OO  ON  THE  SEVENTH  DAY 
OVERDUE. 


UNIVERSITY  OF  CAUFORNIA  LIBRARY 


